For a graph \(\Gamma \)  of order \(n\)  and size \(m\) , let \(d = [d_1, d_2, \cdots , d_n]\)  be a sequence of non-negative integers arranged in non-increasing order, which we refer to as a partition of \(2m\) . Define the transpose of the partition as \(d^* = [d_1^{*}, d_2^{*}, \cdots , d_r^{*}]\) , where \(d_j^{*} = |\lbrace d_i : d_i \ge j\rbrace |\) , \(j = 1, 2,\cdots , r\) . Therefore \(d_j^{*}\)  is the number of \(d_i\) 's that are greater than equal to \(j\) . Recall from [1]} that a sequence \(d^*\)  is called the conjugate sequence of \(d\) . The another interpretation of a conjugate sequence is the Ferrer's diagram (or Young diagram) denoted by \(Y(d)\)  corresponding to \(d_1, d_2, \cdots , d_n\)  consists of \(n\)  left justified rows of boxes, where the \(i^{th}\)  row consists of \(d_i\)  boxes (blocks), \(i = 1, 2, \cdots , n\) . Note that \(d_i^{*}\)  is the number of boxes in the \(i^{th}\)  column of the Young diagram with \(i = 1, 2, \cdots , r\) . An immediate consequence of this observation is that if \(d^*\)  is the conjugate sequence of \(d\) , then,
\(\sum \limits _{i = 1}^{n} d_i = \sum \limits _{i = 1}^{r} d_i^{*}\) 
